Radon integrals. An abstract approach to integration and Riesz representation through function cones
type: Item
instance of: scholarly article
title: Radon integrals. An abstract approach to integration and Riesz representation through function cones
zbMATH Open document ID: 0756.28009
author: Bernd Anger
author: Claude Portenier
published in: Progress in Mathematics
publication date: 1992-09-18T00:00:00Z
review text: A \(\nu:\overline\mathbb{R}^ X\to\mathbb{R}\) is called an upper functional (UF) if \(T:=\{f\in\tilde\mathbb{R}^ X:\nu(f)<\infty\}\) is a lattice cone on which \(\nu\) is positive-homogeneous with \(\nu(s+t)\leq\nu(s\wedge t)+\nu(s\lor t)\leq\nu(s)+\nu(t)\), satisfying \(\nu(f)=\inf\{\nu(t):f\leq t\in T\}\) for \(f\in\overline\mathbb{R}^ X\), with \(\tilde R:=(-\infty,\infty]\), \(X\) a set \(\neq\emptyset\). If \(\mu:S\to\tilde\mathbb{R}\) is positive-homogeneous and additive (=: linear) on a min-closed cone \(S\) from \(\tilde\mathbb{R}^ X\), then \(\mu^*(f):=\inf\{\mu(s):f\leq s\in S\}\) defines an UF. \(f\in\overline\mathbb{R}^ X\) is called \(\nu\)-integrable if \(\nu_ *(f)=\nu(f)\in\mathbb{R}\), with \(\nu_ *(f):=-\nu(-f)\); if \(\nu=\mu^*\), \(f\) is \(\mu^*\)-integrable iff to each \(\varepsilon>0\) there are \(s,t\in S\) with \(-s\leq f\leq t\) and \(\mu(s+t)<\varepsilon\) (abstract proper Riemann integral). For the set \(J(\nu)\) of \(\nu\)-integrable functions the usual linear and lattice theory is obtained, \(J(\nu)\) is closed in \(\overline\mathbb{R}^ X\) with respect to \(d(f,g):=\nu(| f-g|)\). (Though not directly comparable, this \(\nu\)-integral should be subsumed by the Aumann-Schäfke integration with integral metrics [\textit{G. Aumann}, Arch. Math. 3, 441-450 (1952; Zbl 0048.037), \textit{D. Hoffmann} and \textit{F.-W. Schäfke}, ``Integrale.'' Bibl. Inst. Mannheim (1992), \textit{K. Bichteler}, ``Integration theory'' (with special attention to vector measures) (1973; Zbl 0249.28002)].) \(J(\nu)\) is extended to \(J^ 0(\nu):=J(\nu^ 0)\) containing improper absolutely \(\nu\)-integrable functions, with UF \(\nu^ 0(f):=\inf_{u\in J_ -}\sup_{v\in J_ - }\nu((f\wedge(-v))\lor u)\), \(J_ -:=J(\nu)\cap(-\infty,0]^ X\) (which corresponds to Schäfke's local integral norm [\textit{F.-W. Schäfke}, J. Reine Angew. Math. 289, 118-134 (1977; Zbl 0337.28011)]; for the case of finitely additive set functions such an integral has been introduced by the reviewer […
Mathematics Subject Classification ID: 28C05
Mathematics Subject Classification ID: 46G12
Mathematics Subject Classification ID: 28-02
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 28A25
Mathematics Subject Classification ID: 28C15
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 52672
zbMATH Keywords: Riemann integral
zbMATH Keywords: Daniell integral
zbMATH Keywords: integration via cones
zbMATH Keywords: linear functionals as integrals
zbMATH Keywords: upper functional
zbMATH Keywords: Aumann-Schäfke integration
zbMATH Keywords: integral norm
zbMATH Keywords: Dunford-Schwartz integral
zbMATH Keywords: convergence theorem
zbMATH Keywords: completeness
zbMATH Keywords: Radon integral
zbMATH Keywords: Bourbaki integral
zbMATH Keywords: representation theorems
zbMATH Keywords: Daniell-Stone representation theorem
reviewed by: Hans Günzler
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 52672